Question: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $x \neq 0$. $p = \dfrac{x - 6}{x + 5} \times \dfrac{2x + 10}{x^2 - 2x - 24} $
Answer: First factor the quadratic. $p = \dfrac{x - 6}{x + 5} \times \dfrac{2x + 10}{(x - 6)(x + 4)} $ Then factor out any other terms. $p = \dfrac{x - 6}{x + 5} \times \dfrac{2(x + 5)}{(x - 6)(x + 4)} $ Then multiply the two numerators and multiply the two denominators. $p = \dfrac{ (x - 6) \times 2(x + 5) } { (x + 5) \times (x - 6)(x + 4) } $ $p = \dfrac{ 2(x - 6)(x + 5)}{ (x + 5)(x - 6)(x + 4)} $ Notice that $(x + 5)$ and $(x - 6)$ appear in both the numerator and denominator so we can cancel them. $p = \dfrac{ 2\cancel{(x - 6)}(x + 5)}{ (x + 5)\cancel{(x - 6)}(x + 4)} $ We are dividing by $x - 6$ , so $x - 6 \neq 0$ Therefore, $x \neq 6$ $p = \dfrac{ 2\cancel{(x - 6)}\cancel{(x + 5)}}{ \cancel{(x + 5)}\cancel{(x - 6)}(x + 4)} $ We are dividing by $x + 5$ , so $x + 5 \neq 0$ Therefore, $x \neq -5$ $p = \dfrac{2}{x + 4} ; \space x \neq 6 ; \space x \neq -5 $